An improved bi-objective salp swarm algorithm based on decomposition for green scheduling in flexible manufacturing cellular environments with multiple automated guided vehicles

Energy-awareness in the industrial sectors has become a global consensus in recent decades. Green scheduling is acknowledged as an effective weapon to reduce energy consumption in the industrial sectors. Therefore, this paper is devoted to the green scheduling of flexible manufacturing cells (FMC) with auto-guided vehicle transportation, where conflict-free routing of the vehicles is considered. To deal with this problem, a bi-objective optimization model is proposed to achieve the minimization of the maximum completion time and the total energy consumption in an FMC. The studied problem is an extension of flexible job shop problem which is NP-hard. Thus, an improved bi-objective salp swarm algorithm based on decomposition (IMOSSA/D) is proposed and applied to the problem. The approach is based on the decomposition of the bi-objective problem. Salp swarm intelligence along with three stochastic-distribution-based operators are incorporated into the approach, to enhance and balance its exploring and exploiting ability. Computational experiments are performed to compare the proposed approach with two state-of-the-art algorithms. This study allows the decision makers to better trade-off between energy savings and production efficiency in flexible manufacturing cellular environment.


Introduction
The manufacturing societies are pushed to improve their product suits by rapidly changing customer needs, especially during the third industrial revolution (from the 1980s to today) (Yin et al. 2018).The quest for high productivity and low cost is no longer the whole story, while product personalization and on-time delivery are also emphasized.For this purpose, flexible manufacturing has emerged and is growing at a rapid pace which can meet customer needs of high variability and customization with higher flexibility, automation and productivity.It can be implemented in several system configurations, among which flexible manufacturing cells (FMCs) are dominating and particularly favored by small and medium-sized enterprises (Ito 2013).Also, flexible manufacturing systems (FMS) often require the integration of a set of FMCs.FMCs represent not only today's manufacturing industry, but also its future.The Internet of things, electric vehicles, and artificial intelligence, each of which are playing a role in the new industrial revolution, are beginning to be incorporated into FMCs.These technologies will further enhance the strengths of FMCs.
A typical FMC consists of one or more flexible machines, an automated handling system and a central control system (Tuysuz and Kahraman 2010).Processing and transportation are the two main activities of FMCs, which are highly interconnected, making the scheduling of these activities exceedingly complex.Faced with such difficulties, many shop floor scheduling studies choose to focus on processing activities rather than handling processes to achieve simplification.Ignoring the transportation processes, the scheduling of FMC can be simplified as the flexible job shop scheduling problem (FJSSP), an relaxation of the job shop problem (Jurisch 1995).FJSSP is a scheduling problem with important applications and has been widely studied (Deliktas et al. 2021).FJSSPs of their basic version do not involve any transportation process.However, in recent years, more and more researchers have started to emphasize the importance of material handling in shop floor scheduling problems.For example, there are studies on FJSSPs considering transport time (Wang et al. 2021), transportation costs (Ziaee et al. 2022), crane handling (Zhou and Liao 2020), etc. Extensive review on FJSSPs has been done by Gao et al. (2019).
As for transportation in an FMC, it is generally achieved by tools such as conveyors, cranes, and automated guided vehicles (AGV), which can be incorporated into the scheduling model, making the model more realistic.Among these transporting tools, AGVs have been more and more used due to their flexibility, energy-efficiency and cost advantage (Bechtsis et al. 2017).Although material handling with AGV in manufacturing systems is a hot topic in the literature, most studies assume that the path of the AGV is known or the number of the AGVs is unlimited, which is far from the case in reality.For example, Fontes and Homayouni (2019) studied joint production and AGV transport scheduling in flexible manufacturing systems, but with pre-determined travel time between machines.Zhao et al. (2020) designed and implemented a multi-AGV scheduling approach for a job shop, where the AGVs are navigated by magnetic strips.(He et al. 2022) studied an energy-efficient multi-objective production scheduling problem, with which integrated with multiple AGVs with pre-determined routes.
However, the scheduling problem associated with AGVs can be even more complex and closer to real-world production systems if more realistic factors such as vehicle routing, conflict avoidance and battery depletion can be taken into account.Due to its high complexity, too few works have been done so far on conflict-free routing of AGVs in FMCs.Therefore, this paper takes this into consideration, which may bridge some of the gaps between the existing research and the real-world implementations.
The diversity of scheduling criteria needs to be improved in addition to the consideration of the transportation issue in an FMC.Energy efficiency has gradually become another compelling issue nowadays, while operational criteria, such as make span, have been the most focused criterions of shop floor scheduling, (Gao et al. 2020).Reducing energy consumption and greenhouse gas emissions has received widespread attention in manufacturing industry and academia, as global warming and the energy crisis have intensified over the decades (Diaz and Ocampo-Martinez 2019).Due to this reason, green scheduling, also often referred to as energy-efficient scheduling, has been rising as an effective means to promote energy-efficiency in the industrial sector.Green scheduling tries to achieve energy saving through scientific planning of activities in the manufacturing systems, in this case, FMCs.For example, Zhou and Zhu (2021b) worked on multi-objective greening scheduling optimization of part feeding for mixed model assembly lines with mobile robots, and proposed a multi-objective disturbance and repair strategy enhanced cohort intelligence (MDRCI) algorithm.Feng et al. (2020) studied green scheduling of sustainable flexible workshop considering uncertain machine state, and established a hardware system for intelligent monitoring and diagnosis of machinery to gather required data.Chou et al. (2020) designed an energy-aware scheduling algorithm under maximum power consumption constraints, where the energy consumption and production pattern of the machines were modelled in detail.Above all, there is a great opportunity in solving scheduling problem with both operational and energy-efficiency criteria (Zhou and Lei 2021).
Few efforts have been made for energy efficiency in FMCs with AGV transportation, while there have been many studies considering energy costs in many other manufacturing system configurations.The energy consumption of the FMCs, especially the energy consumed during transportation, should be taken into consideration.Also, the transport process and the machining process in FMCs are interlinked and inseparable.Therefore, a comprehensive energy saving model must take both processes into consideration.
This paper proposes and investigates a bi-objective green scheduling problem of flexible manufacturing cells (BOGSP-FMC), considering the transportation process and the energy saving in FMCs.The BOGSP-FMC is an extension of the FJSSP that pursues a trade-off between minimizing energy consumption and make span.Therefore, the problem involved is NP-hard like FJSSP.
In view of the NP-hard nature of the BOGSP-FMC, exact methods including dynamic programming, branch & bound and column generation are difficult to find acceptable solutions in a short time, especially when the problem size is large (Zhou and He 2020;Zhou and Zhu 2021a).To solve such complex multi-objective optimization problems in a reasonable time, many novel intelligent optimization algorithms have been developed and successfully applied to production scheduling problems in the literature, such as multi-objective grey wolf optimizer (Mirjalili et al. 2016), structure enhanced discrete non-dominated sorting genetic algorithm-II (NSGA-II) (Tan et al. 2022) and ant colony optimization (ACO) behavior-based multi-objective evolutionary algorithm based on decomposition (MOEA/ D) (Shao et al. 2022).Multi-objective intelligent algorithms can be divided into at least three categories including dominance-based methods such as NSGA-II (Deb et al. 2002), indicator-based algorithms like IBEA (Zitzler and Ku ¨nzli 2004) and decomposition-based methods such as MOEA/D (Zhang and Hui 2008).Among them, the relatively less applied MOEA/D decomposes the original multi-objective problem into a set of single-objective subproblems with a unique evolutionary mechanism, and its performance has been validated in many complex engineering problems, such as airline cockpit crew rostering optimization (Chutima and Arayikanon 2020) and.
However, MOEA/D still has some drawbacks, such as low local search accuracy and easy to fall into local optimum.Therefore, some improvements and modifications of MOEA/D have emerged in the literature, which can be broadly classified into two categories.The first category is to improve the components of MOEA/D, including collaborative scheme (Zhu et al. 2023), crossover, mutation (Wang et al. 2020), and external archive update methods (Luo et al. 2018).This type of improvements targets one or more components of MOEA/D and can lead to considerable improvements.The second category is the hybridization of MOEA/D with other evolutionary algorithms or swarm intelligence such as PSO (Peng and Zhang 2008), memetic Algorithm (Mashwani and Salhi 2014), and Le ´vy flight (Zhou and Liao 2020).It has been widely acknowledged in the literature that a reasonable synthesis of intelligent algorithms can outperform its parents and many hybrid algorithms have been successfully implemented to solve real-world engineering problems (Barshandeh et al. 2022;Kaya et al. 2021).
Inspired by these two types of improvements, this paper proposes a novel hybrid intelligent algorithm, namely, the improved multi-objective salp swarm algorithm based on decomposition (IMOSSA/D).The proposed algorithm enhances MOEA/D in three aspects.First, a unique crossover competition mechanism is designed to replace the simulated binary crossover (SBX) and improve its search capability.Then, a novel swarm intelligence algorithm, the Salp Swarm Algorithm (SSA) (Mirjalili et al. 2017), with high performance in many engineering problems (Dagal et al. 2022;Yi et al. 2022), is combined with MOEA/D to endow individuals in the population with unique co-evolutionary patterns and more exploitation capabilities.Finally, local search strategies based on stochastic distributions, including Brownian motion and Le ´vy flight (Metzler and Klafter 2000), are added to the algorithm, and their ability in local search and escaping from local optima has been validated by many studies (Balakrishnan et al. 2022;Dong et al. 2021;Gupta and Deep 2019).
This paper is dedicated to apple the proposed IMOSSA/ D to the BOGSP-FMC where a set of jobs await being processed in an FMC.The FMC consists of a set of multifunctional machines, material storage areas, and an AGV material handling system which performs intracellular transfers of jobs.
The contributions of this study are summarized as follows.
(1) The bi-objective green scheduling problem of FMCs is investigated, where material handling via AGVs is considered in detail in addition to machining.The objectives of the problem are to minimize the make span and total energy consumption.(2) A multi-layer solution representation and a corresponding decoding strategy are designed.A conflictfree routing method based on ACO and a direct conflict resolution method is also proposed.(3) A hybrid intelligent algorithm IMOSSA/D is proposed to solve the BOGSP-FMC, which hybridizes MOEA/D and SSA.A crossover competition mechanism is presented to substitute SBX in MOEA/D.Local search strategies namely Brownian motion, Cauchy motion and Le ´vy flight are also incorporated in the method to improve its exploiting ability.
The remainder of this paper is organized as follows.The BOGSP-FMC is depicted in text and pictures and an optimization model is formulated in Sect. 2. Section 3 explains the IMOSSA/D approach in detail.In Sect.4, computational experiments are performed to justify and evaluate the performance of the IMOSSA/D in comparison with the other two state-of-the-art methods.Finally, conclusions and opportunities for future study are discussed in Sect. 5.

Problem description and formulation
An FMC consists of M machines and V identical AGVs.A set of jobs j (j = 1, 2, …, J) are to be processed and transported through the FMC, and job j is composed of K j operations.Operation O k j (k = 1, 2, …, K j ) can be processed on a pre-determined set of machines UðO k j Þ with different process time and process power.
Figure 1 depicts a hypothetical example of an FMC.Each node in the Figure represents an intersection where the AGV can stop or turn, and the location of each node can be denoted to a coordinate.The node (0, 1) is the location of the material storage area and the node (3, 0) is the location of the product storage area.The remaining nodes house machines 1 to machine 6.Among the above nodes, those at which the two storage areas are located allows all vehicles to stop at the same time, while the other nodes allow only one vehicle to stop or pass at the same time.Some nodes are connected by lanes, and AGVs can only travel on the lanes.All lanes can be entered from both ends, but any two AGVs cannot enter a lane from its both ends at the same time.This is often the case due to the limited space in the workshop.The travel speed of AGVs on the lanes can be considered as uniform, and the loading and unloading of jobs are neglected.All the jobs and AGVs are located at material storage area at the beginning, and each job shall be stored in the product storage area after completed.
In the FMC, it is to decide the processing machine and processing sequence for each operation, the AGV designated to each transfer and the transfer sequence, and to plan conflict-free transfer routes.The goal of the BOGSP-FMC is to balance the two objectives of minimizing the total completion time and total energy consumption of the manufacturing cell by making these decisions in a rational manner.

Assumptions
To describe the problem more accurately, the following assumptions are made.

Mathematical model
Based on the problem formulated, a mathematical model is proposed as an accurate depiction of the BOGSP-FMC.
The notations and explanations used in this model are listed in Table 1 and the mathematical model is given as follows.
The addressed problem BOGSP-FMC is bi-objective as illustrated in Function (1).The objective of the is to minimize the make-span and the total energy consumption, as in Objective function ( 2) and (3) respectively.
The energy consumption for processing and transportation are calculated in Eq. ( 4) and ( 5).
Constraint (6) to Constraint (30) are the constraints that the decision variables must obey.Constraint (6) and Constraint (7) make sure that the first operation and the last one processed by each machine are unique.
k Operation ðk ¼ 1; 2; . ..; K j ; K j þ 1Þ, and operation K j þ 1 is dummy in which the job is handled to the product storage area The kth operation of job j m Machine (m = 1, 2, …, M, M ?1), and M ? 1 denotes product storage area The chosen processing machine of O k Power consumption when the AGV is traveling unloaded Power consumption when the AGV is traveling with load The time when The time when O k j is unloaded N s Nodes (s = 0, 1, …, S) in the network of an FMC, in which N 1 denotes the Node where the material storage area locates, N S denotes the product storage area and N sðmÞ denotes machine m

A s
The set of nodes which N s is connected to by lanes Constraint (8) to Constraint (10) ensure each operation of each job is designated to only one machine.
That one machine can only process one operation of one job at the same time is guaranteed by Constraint (11).
Constraint ( 22) ensures a job can only be loaded on an AGV after the immediate predecessor is unloaded.
Constraint (23) and Constraint (24) make sure that a job can only be loaded on the AGV after the precedent operation is done.
Constraint (25) guarantees the unloading time is latter than the loading time of each transportation task.
Constraint (26) and Constraint ( 27) ensure that loading and unloading can only be done at the correct node.
Constraint (28) guarantees the process of each operation can only begin after the job is unloaded.
Constraint (29) makes sure the process is nonpreemptive.
Constraint (30) declares that some of the variables can only be zero or one.
3 Improved multi-objective salp swarm algorithm based on decomposition The BOGSP-FMC studied in this paper is an extension of the flexible job shop scheduling problem, so it is NP-hard and most suitable to be solved by meta-heuristics.So, an improved multi-objective salp swarm algorithm based on decomposition (IMOSSA/D) is proposed, which is based on the hybridization of MOEA/D and an improved salp swarm algorithm (SSA).First, the initial population is generated randomly.The crossover operator in MOEA/D is substituted by three crossover operators organized by a competition mechanism.Also, three mutation operators based on stochastic distribution, namely Brownian motion, Cauchy motion and Le ´vy flight are randomly chosen after the SSA to perform local search of the individuals in the population.The framework of the proposed algorithm is shown in Fig. 2. The proposed approach is presented in four parts in this section.First, a real number-based solution representation is proposed and illustrated by a numerical example.Then, a conflict-free routing method based on ACO is presented, in which a conflict avoidance mechanism is designed.Thereafter, a repair mechanism is introduced to fix any infeasible solutions that arise during the evolution of the algorithm.Finally, the composition of IMOSSA/D is described in detail, and the computational complexity of IMOSSA/D is analyzed.

Solution representation
Careful design of encoding and decoding methods is required when applying metaheuristics to solve specific discrete optimization problems.A three-layer real-numberbased solution representation is proposed, each layer of which represents the processing order, machine assignment, and the assignment of AGVs respectively.
All the three layers of a solution is a vector of P j K j elements, each of which corresponds to one operation of one job.The elements of the first layer are real numbers which belongs to ½0; 1, indicating the priority of their operations.The smaller the number, the higher the priority.In the second layer are real numbers belonging to ½0; jUðO k j Þj, and a number which belongs to ½0; 1Þ represents that machine 1 is assigned, and so on.The third layer consists of real numbers between ½0; V for assigning AGVs.
The decoding procedures are to interpreting the solution representation and to calculate the two objectives.The details of them are listed as follows.
Step 1 Obtain the processing order of each operation from the first layer of the solution.
Step 2 Allocate machines to operations and record the processing sequence of machines.
Step 3 Allocate AGVs to operations and record the transporting sequence of AGVs.When two adjacent operations of the same job are processed on the same machine, the latter operation will not require AGV handling.
Step 4 Identify the starting, loading and unloading nodes of each AGV.
Step 5 From t = 0, start the process and handling in FMCs and identify the state of jobs (operations processed), machines (tasks performed) and AGVs (tasks performed and locations et al.) of each timestep.These states can only and must be changed when the required constraints are met.
To show the decoding process more clearly, an example is given for illustration.As shown in Fig. 3, 3 jobs are involved, each of which contains 2 operations.The jobs are processed by 3 machines and handled by 2 identical AGVs, and the 3 machines can process any operation of any job (for simplification).Figure 3a gives the representation of a feasible solution.Figure 3b performs a continuous-discrete-space transformation, converting the real numbers into processing priority and indexes of machines/AGVs. Figure 3c gives the decoding procedures based on step 1 to step 4. In the handling route presented in step 4, ) denotes a loaded route and !denotes an unloaded route.It is worth noting that the AGV indexes in Fig. 3b highlighted by a circular box are two adjacent operations of the same job Fig. 2 The framework of the IMOSSA/D which are processed on the same machine.So, the operation O 2 3 does not require handling.

Conflict-free routing method
By decoding steps 1-4, the starting node, end node and load of each AGV for each handling task are obtained, meeting the prerequisites for routing.The proposed routing method is based on the ACO (Dorigo et al. 1996), the main steps of which are as follows.At time step t, if the AGV is located at point ðx; yÞ, it can change to five optional states (xþ; xÀ; yþ; y À andstay) at moment t þ 1.Each one is assigned a certain state transfer probability, and the action of each AGV between time step t and t þ 1 is determined randomly according to the above probabilities until it arrives at the end node.The ACA is designed to gradually optimize the state transfer probability and thus to find a shorter path.
To decide the action of each AGV between time step t to t þ 1, whether it is close to the end node is the first concern, and then, possible route conflicts must be considered.Under certain circumstances, an AGV choosing the best direction may force other AGVs to take a detour to avoid conflicts.The proposed ACA copes with the above problem by introducing randomness and pheromone mechanism.
Specifically, state transfer probability p d xy ðl; s; aÞ denotes the probability that an AGV chooses state d at point (x, y) in the lth move.where s denotes the number of stages of the ACA and a is the index of ants.It is calculated as follows.
where g d xy l ð Þ is the heuristic function, obtained from Eq. (32), and s d xy l; s ð Þ denotes the pheromone concentration Fig. 3 An example of decoding procedures of the AGV selecting state d at the point (x, y).When all ants have completed the pathfinding, p d xy l; s; a ð Þ must be updated according to Eq. ( 32), ( 33) and (34).
In Eq. ( 34), the initial value of Ds d xy ðl; s; aÞ is 0, which will increase Q Tðl;s;aÞ each time ant a selects that path (action).Tðl; s; aÞ denotes the total time consumed by ant a in that handling task.
A conflict elimination mechanism is designed for path conflicts of multiple AGVs.Three types of conflicts may occur among AGVs as shown in Fig. 4. When the current time step is t, the steps of path planning at time t þ 1 are as follows.
Step 1 Determine the priority order of path selection of each AGV.
Step 2 Determine the action at time t þ 1 of the unacted AGV of the highest priority randomly according to p d xy ðl; s; aÞ, and any conflicts with acted AGVs must be avoided.
Step 3 Check the state selection probability of each travel direction of the unacted AGVs.If there is an AGV with state selection probability of 0 for each direction, withdraw the action of the AGV in step 2 and prohibit the AGV from selecting that action again.If there is unacted AGVs left, go back to step 2, otherwise end path selection.

Infeasible solution repair mechanism
Infeasible solutions may be produced during the iterations of the IMOSSA/D, which can be repaired to be feasible.There are two types of infeasibility that can be generated.First, the elements of a solution may surpass the prescribed upper or the lower bound, which can be repaired by randomly regenerating another real number in the predetermined range.Secondly, the processing sequence of the operations of the same job may be violated.Infeasibility of this type can be repaired by rearrange the processing priority of the same job in the first layer of the infeasible solution in ascending order.These repair mechanisms can preserve the information contained in the infeasible solutions to the maximum extent and transform them into feasible solutions.
The repair mechanisms are illustrated with the example introduced in 3.1.An example of infeasible solution before and after repair is shown in Fig. 5a, b respectively.First, the blue ellipse identifies the first type of infeasibility and its repair.The infeasible element in the three layers of the solution are out of bounds ½0; 1, ½0; 3 and ½0; 2, respectively, and the repair process is to regenerate the values randomly within the respective range.Then, the yellow box identifies the second type of infeasibility and its repair: the processing priority of the two operations of Job 2 violate the constraint, thus they are rearranged in ascending order to repair.

Population initialization
The IMOSSA/D is designed based on MOEA/D, which, like other variants of MOEA/D, needs to transform the multi-objective optimization problem into a set of scalar subproblems with a certain decomposition method.The Chebyshev decomposition is employed, and thus the subproblems obtained by the decomposition can be defined as follows.
where z Ã denotes the reference point, m is the number of objectives, k i is the weight vector assigned to each subproblems.
Based on this decomposition method, the initial population of the IMOSSA/D is generated by the procedures as follows.
Step 1 Randomly initialize a set of weight vectors k 1 ; k 2 ; :::; k i ; :::; k N , where N is the population size.Initialize the external archive EA ¼ £.
Step 2 For each weight vector k i , calculate the Euclidean distance between it and the rest of the weight vectors.And find the T weight vectors with the closest distance corresponding to T neighborhood solutions.
Step 3 Randomly generate N initial solutions x 1 ; x 2 ; :::; x i :::; x N , each of which corresponds to one subproblem.Calculate the m objectives of each solution.
Step 4 Initialize the reference point z Ã , z The objectives of each subproblem, F x i ð Þ; 1 i N, are obtained by solving Eq. (35).

Competition mechanism of crossover operators
The crossover operator in MOEA/D randomly selects two solutions and performs a simulated binary crossover (SBX), and the child solutions will replace the parent solutions if they are better.This crossover operator allows the solutions of subproblems to exchange information with each other during the evolution and is the core operator of MOEA/D.It has been shown that the use of multiple operators is better for maintaining the diversity of the population than the use of a single SBX (Wang et al. 2020), avoiding the cliff decline of diversity in the late iterations.
Inspired by this, this paper proposes the competition mechanism of multiple crossover operators.
First, uniform crossover (UX), two-point crossover (TPX) and SBX, which have superior performance and large differences, are selected as alternative operators.For each crossover operator, one crossover operator X i ; i ¼ 1; 2; I is chosen randomly according to the selection probability.
where C i t is the contribution of X i in the last t iterations.The contribution is defined as the number of times the parent solution is replaced by the children generated by the operator.

Improved salp swarm algorithm
SSA is a swarm-intelligence-based meta-heuristic proposed by Mirjalili et al. (2017).The metaheuristic, inspired by the behavior of salp swarms in nature, not only performs well in solving the benchmark problems, but also has been applied by researchers to optimization problems such as electric power dispatch (Kansal and Dhillon 2020), feature The salp swarm in nature spontaneously form ''chains'' to find food sources.SSA divides the entire population into one leader and the followers.The leader is the individual leading the swarm at the front of the chain, while the other individuals follow each other one by one.Like other swarm intelligence algorithms, the position of a salp within the search space is determined by a n-dimensional vector.It is also assumed that there is a food source F in the search space as the target of the swarm.The location of the leader is updated according to Eq. ( 36).
where, x 1 j is the coordinate of the position of the first salp (leader) in the jth dimension.F j is the jth dimension of coordinate of the food source.ub j and lb j are the upper and the lower bound of the jth decision variable and c 1 , c 2 and c 3 are random numbers.
Equation ( 36) indicates that the leader salp update its position only according to the food source, which is the best individual of the previous generation population.c 1 is made adaptive with iterations as in Eq. ( 37) while c 2 and c 3 are uniform random numbers between ½0; 1.
where l is the current number of iteration and L is the maximum one.
The positions of the followers are updated as Eq. ( 38).
SSA is modified in three ways as follows to adapt to the BOGSP-FMC with two objectives, decision variables of high dimension and complex search space.
(1) Food source updating In the original single-objective SSA, the food source is the best optimal solution of the previous generation population, which must be modified to solve a multi-objective problem.The best solutions chosen from the historical populations are saved into the EA.Thus, the food source is a solution selected from EA with roulette method.
(2) Updating the position of the followers The updating equation for the followers in original SSA enables full ability of exploration in the earlier stage of the evolution.But the exploitation of the followers is rather less, which can make them abandon some promising area in the search space, especially for a multi-objective one.In the same time, the followers have no memory about their historical positions, which are underutilized.Hence Eq. ( 38) is modified as Eq. ( 39).
where c 4 is the memory coefficient, c 5 denotes the  X where X i is the initial solution and X i;new b is the child one, a b is the step size coefficient, È denotes multiplying by elements, and B obey the standard normal distribution.
The expression of Cauchy motion is as follows.
where C obey the Cauchy distribution, the distribution function of which is Eq. ( 44).
where g ¼ 1 is the proportional coefficient and y obeys the standard uniform distribution.
The expression of Le ´vy flight is presented in Eq. ( 45).
Fig. 6 Experimental results of 30 small sized problem instances where q $ Normalð0; r 2 q Þ and r $ Normalð0; r 2 r Þ, and these coefficients are calculated as follows.

External archive maintenance with quality indicator
An EA is introduced to document best solutions generated during the iterations of the metaheuristic, which may not be preserved otherwise.An EA is usually updated with nondominated sorting and crowding degree indicator.Binaryindicators are promising to compare two solution sets, with lower computational cost and better reflection of dominance relationships.The e-quality indicator proposed by Luo et al. (2018) is utilized to update EA.
Let population P be a sample in the decision space X, and the fitness F I ðxÞ of an individual x 2 P is defined in Eq. ( 48), ( 49) and (50).
The EA is updated by steps as follows.
Step 1 Combine EA and the current generation P to get S ¼ EA [ P, Retain only the non-dominant solutions in S by non-dominant sorting.
Step 2 If number of the solutions in S ( S j j) is no more than N EA , Make EA ¼ S and the updating is finished, otherwise go to Step 3. Step 3 Eliminate one solution with the smallest F I ðxÞ at a time, until S j j ¼ N EA .

Computational complexity analysis
Let the population size be N, the number of objectives be m, and the number of the decision variables be D, then the computational complexity of each ingredient of the IMOSSA/D is listed in Table 2.As can be seen in the table, the computational complexity of the proposed approach is OðN 2 þ NDÞ.

Computational experiments
To evaluate the performance of the IMOSSA/D proposed in this paper, it is implemented with Python to solve a set of test problem instances with different features.First, the way these test problems are generated is described in detail.Then, a set of performance indicators are introduced to evaluate the multi-objective algorithm.Next, evaluated with the indicators, the proposed algorithm is compared with two state-of-the-art intelligent algorithms on FJSSP, MOGWO (Luo et al. 2019) and PLMEAPS (Zhou and Liao 2020).Thereafter, the effects of the major ingredients of the IMOSSA/D are verified experimentally.

Problem instances and parameters
The problem instances are generated randomly, since there are no benchmark problems in the literature.The locations of the machines and storage areas are random.The number of operations of each job in one instance is designated from discrete uniform distribution DU½1; 5.The process time of each operation is generated from U½10; 50, and the process power of each machine from U½10; 20.The transfer power of any AGV is 8 if empty or 15 otherwise.Each problem instance can be represented by a combination of codes J, M and V representing the number of jobs, the number of machines and the number of AGVs, respectively.For instance, J20M5V2 represents an instance involving 20 jobs, 5 machines and 2 AGVs.Inspired by Zhou and Liao (2020), these instances are divided into three groups including small (J 100), medium (100 J 200) and large ones (J [ 200) by the number of jobs involved.
The three involved algorithms are executed 15 times on each problem instances with different random seeds, on a PC with Intel Core i5 2.9 GHz CPU and 16 GB RAM.

Performance indictors
A set of unary indicators are introduced to evaluate the involved algorithms including the degree of pareto optimality (DPO), generational distance (GD), diversity (D) and hyper volume (HV) (Zhou and Zhu 2021b).
1:1:1.DPO measures the number of pareto solutions obtained by one algorithm, defined as follows.
where PF is the pareto front obtained by the involved algorithm while PF Ã denotes the true pareto front.
2:2:2.GD indicates the convergence degree of solution sets, denoted as follows.
where D i is the Euclidean distance between the ith solution in PF and PF Ã .3:3:3.D is the measure of diversity and evenness, and a smaller D indicates a solution set with more desirable distribution, defined as follows.
where d i is the Euclidean distance between the ith solution in PF and the others, d denotes the mean value of d i , d e j is the Euclidean distance between the extreme solution of the jth dimension of PF and PF Ã , and m denotes the number of objectives.4:4:4.HV is the volume of the hyper-polyhedron enclosed by the non-dominated solutions and the reference point in the search space, measuring the degree of dispersion and convergence of the solution sets, defined in Eq. ( 54).
where d represents the Lebrun measure and v i is the volume of the hypercube enclosed by the ith solution and the reference point.

Parameter tuning
The parameters of the algorithms are tuned with Taguchi method (Zhou and He 2021), which is a widely used method for deciding the values of multiple parameters.For problem of different size, different values are chosen by the Taguchi method.The parameter settings for the approaches involved in the experiment are shown in Table 3.

Experimentation and analysis of results
Ninety problem instances are designed according to 4.1 to evaluate the performance of the proposed method and the two benchmark methods, where 5 machines and 1 to 3 vehicles are involved.The performance of the three algorithms on problem instances of different sizes is analyzed by dividing all problem instances into three groups according to their size, i.e., the number of jobs involved.
The instances involving 10 to 100 jobs are small instances, those involving 110 to 200 jobs are medium sized ones and those containing 210 to 300 jobs are divided into largescale group.The performance of the three algorithms on otherwise Flag = 0. Best values for each indicator are bolded in the Table 5.
In Table 5, it can be seen that C max and TE of M are the smallest most of the time.In view of indicator Flag, it takes the value of ? 1 for 21 times out of the 30 instances.HV of M takes the largest value for 28 times, and usually maintains this advantage from earlier iterations, as seen by indicator p.

Effect of the stochastic-distribution-based operators
The justification of stochastic-distribution-based operators is performed in the same way as that of the crossover operators.MOEA/D incorporated with all the three operators is denoted as LS, and those with Brownian motion, Cauchy motion and Le ´vy flight are BM, CM and LF respectively.LS, BM, CM, LF and D are run independently fifteen times for the problem instances and the results is shown in Fig. 10.
LS notably outperforms the other four algorithms in DPO, and is also the best in GD for most of the instances.However, LS has shown little advantage in D over the other methods.

Managerial applications
To bridge some gaps between the proposed approach and its real-world implementation, this section is to put some insights in its managerial applications.
The bi-objective approach pursues the tradeoff between two objectives and may yield multiple non-dominant solutions.Then, it is for the decision makers to choose from the solutions.Different decision makers may have different opinions on the importance of the two objectives, namely the make span and the energy cost.Thus, a preference vector l ¼ ðl 1 ; 1 À l 1 Þ, is used to describe the relative attention given by decision makers to the two objective functions.
The decision maker can take the value of l 1 based on the relative importance of f 1 and f 2 .The importance of f 1 is positively correlated with the value of l 1 .In this paper, the value of l 1 is equally divided into 5 intervals, namely [0, 0.2), [0.2,0.4),[0.4,0.6), [0.6, 0.8), [0.8, 1], each of which corresponds to an applicational scenario.As an illustrative example, a medium-scale problem instance J150M5V2 is solved and the results is presented in Fig. 11.
As shown in Fig. 11, the pareto solution set is divided into 5 groups according to different intervals of preference vector and the number of alternatives left for decision makers is narrowed down.
In sight of the managerial applications of the addressed problem in this section, we believe that this study can provide the managers with an approach to create greener flexible manufacturing cells with guaranteed productivity, making their enterprise more competitive and sustainable.

Case study
A manufacturer of complex workpieces produces three main types of products: fixed frames, hydraulic servo machine access bodies and fuse holder housings.Machining is completed using a flexible manufacturing cell, which contains 17 processing equipment and 2 handling robots.The processing equipment includes 3 lathes, 9 milling machines, 2 boring machines, 2 heat treatment machines and 1 plating machine.There is a batch of workpieces to be processed and the raw materials and all equipment are ready.This batch of workpieces contains information on workpiece type, batch size, process flow, and processing time as shown in Table 6.The equipment information such as type and energy consumption of each processing equipment is shown in Table 7.
The proposed algorithm and the benchmark algorithms are used to solve the above case, and the Pareto front obtained by each algorithm is shown in Fig. 12.It can be seen that the performance of IMOSSA/D is relatively excellent and can be well used for the solution of realworld problems.

Conclusions and future research
Inspired by the opportunity in introducing energy-efficient criterion and conflict-free routing for AGVs in FMCs, this paper models a bi-objective green scheduling problem for flexible manufacturing cells.A realistic consideration of AGV routing can bridge some gaps between research and engineering practice.At the same time, energy costs incurred during processing and transportation, which are complexly interlinked, is calculated in a more simplified way.To solve this problem, we designed an approach based on MOEA/D.The approach absorbs three strategies to adapt the MOEA/D to the problem.First, the competition mechanism of crossover operators substitutes the SBX in MOEA/D to enhance its search ability.Then, an improved salp swarm algorithm and three stochastic-distribution-based operators are also incorporated.At last, an external archive update method with quality indicator is introduced.Computational experiments are performed to evaluate the proposed algorithm on the studied problem, and it outperforms MOGWO and PLMEAPS.The three components of the IMOSSA/D are also validated experimentally.
Future study may involve other greening practice such as shutdown reboot policies and processing speed tuning.In addition, more practical consideration, such as battery depletion of AGVs and machine downtime, can be introduced.As for the approach, the proposed meta-heuristic still has some weaknesses in terms of solution diversity.Exact methods are also needed for smaller version of the studied problem.

( 1 )
Each machine and AGV can process or handle only one job at a time, and each job can be processed by only one machine or handled by one AGV at a time.(2) If two consecutive operations of a job are processed on the same machine, the job does not need to be handled by an AGV between the two operations.(3) The AGVs travel on the lanes with uniform speed.(4) The setup times of the machines are not considered.(5) The loading and unloading time of the AGV is ignored or incorporated in the travel time.(6) Recharge of AGVs and equipment failure are not considered, and the AGVs and machines are always available throughout the manufacturing process.

Fig. 4
Fig. 4 Three types of routing conflicts Brownian motion, Cauchy motion and Le ´vy flight are three main stochastic-distribution-based methods, all of which are now widely used in the field of intelligent optimization.Among these three operators, Brownian motion has a smaller step size and stronger local search ability; Cauchy motion has a larger step size, which can improve the global search ability of the algorithm; and Le ´vy flight has small steps mixed with occasional large steps, which can effectively help avoid premature of the algorithm.A combination of the three operators is added to the IMOSSA/D, which can take advantage of all three at the same time.The mathematical expression of Brownian motion is as follows.

Fig. 7
Fig. 7 Experimental results of 30 medium sized problem instances

Fig. 8
Fig. 8 Experimental results of 30 large sized problem instances

Fig. 10
Fig. 10 Performance of RW, BM, CM, LF and D

Table 1
Notations and explanations

Table 4
Paired T-tests on the performance of the IMOSSA/D versus the benchmark methodsThe improved salp swarm algorithm is denoted as M, while MOEA/D hybridized with SSA is S and MOEA/D is D. These three algorithms run independently fifteen times for problem instances of different scale.The experimental results are listed in Table5, where C max and TE denote normalized mean maximum completion time and mean total energy consumption of the pareto front.Indicator p denotes the average number of iterations (rounded up) where HV of M starts to be less than those of the other two algorithms.Flag indicates the dominance state of M, and if C max and TE of M are both smaller than those of S and D, then Flag = ?1; if they are both larger, then Flag = -1;

Table 5
Performance of D, M and S

Table 7
The information of the equipment